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Stable perfectly matched layers for a class of anisotropic dispersive models. Part II: Energy estimates

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HAL Id: hal-01419682

https://hal.inria.fr/hal-01419682v2

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dispersive models. Part II: Energy estimates

Maryna Kachanovska

To cite this version:

Maryna Kachanovska. Stable perfectly matched layers for a class of anisotropic dispersive models.

Part II: Energy estimates. 2017. �hal-01419682v2�

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DISPERSIVE MODELS. PART II: ENERGY ESTIMATES

Maryna Kachanovska

1

Abstract. This article continues the stability analysis of the generalized perfectly matched layers for 2D anisotropic dispersive models studied in Part I of the work. We obtain explicit energy estimates for the PML system in the time domain, by making use of the ideas stemming from the analysis of the associated sesquilinear form in the Laplace domain. This analysis is based on the introduction of a particular set of auxiliary unknowns related to the PML, which simplifies the derivation of the energy estimates for the resulting system. For 2D dispersive systems, our analysis allows to demonstrate the stability of the PML system for a constant absorption parameter. For 1D dispersive systems, we show the stability of the PMLs with a non-constant absorption parameter.

1991 Mathematics Subject Classification. 65M12, 35Q60.

.

Introduction

The method of the perfectly matched layers (PML), introduced by B´erenger [1, 2], is used ubiquitously in engineering and physics communities to compute a solution to a problem posed in an unbounded domain. However, it is well-known that for some classes of problems (e.g. the wave propagation in anisotropic and/or dispersive media) the classical PML method may result in instabilities [3,4]. For a class of anisotropic dispersive models PMLs were stabilized [5].

In this work we continue the analysis of the stable PMLs constructed in [5]. This class of systems describes the wave propagation in metamaterials and plasmas in 2D. In the frequency domain, they can be written in the form of an anisotropic dispersive wave equation:

ε1(ω)−1∂x2u + ε2(ω)−1∂y2u + ω2µ(ω)u = 0, ω ∈ R,

where ε1(ω), ε2(ω) have a meaning of a dielectric permittivity and µ(ω) is a magnetic permeability. They

depend on frequency non-trivially and satisfy Im(ωεj(ω)) > 0, j = 1, 2, Im(ωµ(ω)) > 0 for Im ω > 0, an

assumption connected to the stability of the model, cf. [6].

In the time domain, such systems correspond to the Maxwell’s equations with currents, which are coupled to the electromagnetic field through the ordinary differential equations.

Keywords and phrases: perfectly matched layers, Maxwell equations, Laplace transform, energy, Fourier analysis

This work was supported by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH,

LabEx LMH, as well as a co-financing program PRESTIGE

1 Laboratoire Poems (UMR 7231 CNRS/Inria/ENSTA ParisTech), ENSTA ParisTech, 828 Boulevard des Mar´echaux 91120

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In [5] a method of construction of stable perfectly matched layers for this class of models is suggested. It is based on the ideas of B´ecache et al. [4] for isotropic metamaterials, which, in turn, generalizes the work of Cummer [7] for the Drude model. However, the analysis in the aforementioned works does not cover the question of the decay of an energy associated with the newly constructed PML systems.

This questions was answered for the classical PML applied to isotropic, non-dispersive Maxwell equations by B´ecache and Joly [8]. In a more general case, the derivation of energy estimates for PML systems does not seem trivial. For example, in [8] the authors have obtained energy estimates for the PML for isotropic Maxwell equations written in the Zhao-Cangellaris formulation [9]. The derivation of the energy estimates for this formulation is simpler compared to e.g. B´erenger’s split formulation. Our approach is different from that of [8], where the PML system was given a priori. We suggest to look for a formulation that would be equivalent to the original system and would allow to obtain the energy in a simpler way. We make use of the stability analysis in the Laplace domain, as it was done in the work [5], which provides intuition on derivation of explicit (without appeal to the Plancherel’s theorem) energy estimates.

An alternative way to derive energy estimates is offered by the use of the method of Hagstrom and Appel¨o [10]. It can be applied to a very general class of systems, however, it results in energy estimates on the spatial derivatives of unknowns [11], while most of the estimates we obtain involve the original unknowns. Moreover, the application of this approach to dispersive models does not seem trivial, due to the need to rewrite the original problem as a single equation, and then apply the method of [10] to the resulting problem.

This work is organized as follows. In Section 1 we state the problem under consideration. In Section 2 we show how to construct stable PMLs for so-called generalized Lorentz models and derive the energy estimates for the corresponding time-domain system. In Section 3 we extend these results to general passive materials. Finally, in Section 4 we demonstrate how the ideas of the previous sections can be used to prove the stability of the PMLs of [5] for 1D dispersive systems, for a non-constant absorption PML parameter.

1. Problem Setting

The wave propagation in 2D dispersive media is described by the Maxwell system: ∂tDx= ∂yHz,

∂tDy= −∂xHz,

∂tBz= −∂xEy+ ∂yEx.

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In the above we use the scaling c = ε0 = µ0 = 1. The relations between the electric field and the electric

displacement field, as well as between the magnetic field and the magnetizing field are given in the Laplace domain. Denoting by s the Laplace variable, we set ˆu(s) = (Lu) (s) the Laplace transform of u(t) and introduce C+= {s ∈ C : Re s > 0}. With this notation, the constitutive relations read ˆD = ε(s) ˆE, ˆHz= µ−1(s) ˆBz. We

will assume that the dielectric permittivity is a diagonal matrix

ε(s) =εx(s) 0 0 εy(s)

 .

The multiplication in the Laplace domain becomes a convolution in the time domain, which we formally denote D = ε(∂t)E, Bz= µ(∂t)Hz.

The system (1) in the Laplace domain can be rewritten as an anisotropic dispersive wave equation:

s2µ(s) ˆHz− εy(s)−1∂x2Hˆz− εx(s)−1∂y2Hˆz= 0. (2)

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Definition 1.1. A function c : C+→ C is passive if it is analytic in C+ and satisfies Re (sc(s)) > 0 there.

In this work we will assume that εx, εy, µ are passive. One of the examples of passive models is offered by

so-called generalized Lorentz materials [12]:

εα(s) = 1 + nα X `=0 εα` s2+ ω2 α` , εα`> 0, ωα`∈ R, ` = 0, . . . , nα, α ∈ {x, y}, µ(s) = 1 + nµ X `=0 µ` s2+ ω2 µ` , µ`> 0, ωµ`∈ R, ` = 0, . . . , nµ. (3)

The generalized Lorentz materials include Drude materials [4] and a 2D uniaxial cold plasma [12, 13].

Remark 1.2. The passivity assumption covers nonhyperbolic models as well (e.g. the diffusion equation, with x(s) = y(s) = 1, µ(s) = s−1).

In [5] it was demonstrated that the well-posedness and stability of (1) follows from the passivity of coefficients of the sesquilinear form associated with (2). Introducing the scalar product

(u, v) = Z

R2

u¯vdx, u, v ∈ L2,

we can define the following class of sesquilinear forms. Definition 1.3. We will call a sesquilinear form

A(u, v) = a(s)(∂xu, ∂xv) + b(s)(∂yu, ∂yv) + s2c(s)(u, v), u, v ∈ H1(R2),

passive if a(s)−1, b(s)−1, c(s) are passive. Informally,

Passivity of a Sesquilinear Form =⇒ Time-Domain Stability of the Corresponding System. (4) Instead of resorting to the Laplace-domain analysis of [5], one can prove stability results directly in the time domain, using energy techniques, cf. [14]. Extending these results to the PMLs is the subject of present work.

2. Stability of PMLs for Lorentz Materials

The analysis for the generalized Lorentz materials (3) will play a crucial role in this work. This is due to the fact that any passive material can be approximated by generalized Lorentz materials, see Section 3. Moreover, Lorentz materials are the only representatives of a fairly general class of passive materials. To demonstrate this, let us assume that εx, εy, µ satisfy the following assumptions, justified in [4, 12, 14].

Assumption 2.1. A function r(s) is a rational function r(s) = 1 +pr(s2)

qr(s2), where the polynomials pr and qr

have real coefficients, no common roots, and also deg pr< deg qr. All its zeros and poles lie in iR.

A class of passive functions satisfying the above condition is quite narrow, see [14] and references therein. Theorem 2.2. A function r(s) satisfying Assumption 2.1 is passive if and only if r(s) = 1 +

n P `=0 r` s22 ` , where ω`∈ R and r`> 0 for all ` = 0, . . . , n.

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Thus, passive functions that satisfy Assumption 2.1 correspond to Lorentz materials (3).

This section is organized as follows. In Section 2.1 we formulate the time-domain system corresponding to Lorentz materials without the PML. Next, in Section 2.2 we discuss the construction of stable PMLs for dispersive systems, concentrating in particular on Lorentz models. In Section 2.3 we show the main idea of the derivation of the PML system that would allow to obtain simplified energy estimates. Finally, in Sections 2.4 and 2.5 we derive the energy estimates for systems corresponding to different PMLs.

2.1. Time-Domain System without PMLs

We consider the Maxwell system (1) with Lorentz parameters (3). In the time domain, we obtain the Maxwell equations with currents, coupled to E, Hz via ODEs, cf. [4]:

∂tDx= ∂yHz, (5a) ∂tDy= −∂xHz, (5b) ∂tBz= ∂yEx− ∂xEy, (5c) ∂tDα= ∂tEα+ nα X `=0 εα`λα`, (5d) ∂tλα`+ ωα`2 pα`= Eα, ∂tpα`= λα`, ` = 0, . . . nα, α ∈ {x, y}, (5e) ∂tBz= ∂tHz+ nµ X `=0 µ`λµ`, (5f) ∂tλµ`+ ω2µ`pµ`= Hz, ∂tpµ`= λµ`, ` = 0, . . . nµ. (5g)

Let us clarify the time-domain realization of ˆBz = µ(s) ˆHz (the time-domain equivalent of D = ε(s)E can be

obtained similarly). The latter equivalently reads s ˆBz= sµ(s) ˆHz=

 s + nµ P `=0 µ` s+ω2 µ`s−1  ˆ Hz. One defines ˆ λµ` = 1 s + ω2 µ`s−1 ˆ Hz, or, equivalently, sˆλµ`+ ω2µ`pˆµ` = ˆHz, sˆpµ`= ˆλµ`.

Provided that λµ`|t=0= 0 = pµ`|t=0, the above coincides with (5g) in the time domain. Similarly, for Bz|t=0=

Hz|t=0, we verify that (5f) coincides in the Laplace domain with ˆBz = µ(s) ˆHz. Let us formulate the following

known result for the system (5a-5g), see also [4]. We provide its proof since it will be of use later. Theorem 2.3. An energy E of (5a-5g), defined below, satisfies d

dtE = 0. Here E = 1 2 kExk 2+ kE yk2+ kHzk2 + Ex+ Ey+ Ez, Eα= 1 2 nα X `=0 εα` kλα`k2+ ωα`2 kpα`k2 , α ∈ {x, y}, Ez= 1 2 nµ X `=0 µ` kλµ`k2+ ωµ`2 kpµ`k2 .

Proof. Test the equation (5c) with Hz, to obtain

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We consider the first term of the above expression; using (5f), (∂tBz, Hz) = 1 2 d dtkHzk 2+ nµ X `=0 µ`(λµ`, Hz) (5g) = 1 2 d dtkHzk 2+ nµ X `=0 µ` λµ`, ∂tλµ`+ ω2µ`pµ` (5g) = 1 2 d dtkHzk 2+1 2 d dt nµ X `=0 µ`kλµ`k2+ nµ X `=0 µ`ω2µ`(∂tpµ`, pµ`) = 1 2 d dtkHzk 2+ d dtEz. (7) The term (Ex, ∂yHz) in (6) can be rewritten exactly like in the previous case:

(Ex, ∂yHz) (5a) = (Ex, ∂tDx) (5d) = 1 2 d dtkExk 2 +1 2 d dt nµ X `=0 ε` kλx`k2+ ω2x`kpx`k2 = 1 2 d dtkExk 2 + d dtEx. (8)

Similarly one shows that −(Ey, ∂xHz) = 12dtdkEyk2+dtdEy. 

2.2. Construction of Stable PMLs for Dispersive Models

Consider the system (1) written in the Laplace domain (2). To construct a PML in the region x ≥ 0, one assumes that (2) is valid for x = ˜x ∈ C. The corresponding analytic continuation of ˆHz, denoted by ˆHz, solves

(2), with x = ˜x. In [4] it is proposed to choose ˜x as follows, for some analytic function ψ(s),

˜ x =        x + s−1ψx(s) x Z 0 σx(x0)dx0, x ≥ 0, x, x < 0, σx(x) =    σ(x) ≥ 0, x ≥ 0, 0, x < 0. (9)

The classical PML can be obtained taking ψx(s) = 1. For dispersive materials this choice may be unstable. We

discuss how to choose ψx(s) in further sections. With this change of variables, (2) transforms into

ε−1y 1 + σxs−1ψx −1 ∂x  1 + σxs−1ψx −1 ∂xHˆz  + ε−1x ∂yyHˆz− s2µ ˆHz= 0, x ∈ R. (10)

By analytic continuation, ˆHz(x, y) = ˆHz(x, y) for x < 0. The resulting system should be rewritten in the time

domain. We postpone this question to future sections. The PML in other directions (e.g. for y ≥ 0) can be constructed similarly. In corners the changes of both of the variables x and y is used.

Under (mild) assumptions on the coefficients of the sesquilinear form, one can show that the time-domain system corresponding to (10) is well-posed, for any σx(x) ∈ L∞(R), see [15]. It seems significantly more difficult

to show the stability of the corresponding problem for variable σx(x), cf. [16]. Finer results can obtained by

studying the sesquilinear form corresponding to (10) with σx(x) = const, x ∈ R, when (9) becomes, cf. [5],

x → x 1 + s−1ψxσx , σx≡ const ≥ 0, x ∈ R. (11)

In this case the sesquilinear form corresponding to (10) becomes

A(u, v) = εy(s)

−1

(1 + s−1ψ

x(s)σx)2

(∂xu, ∂xv) + εx(s)−1(∂yu, ∂yv) + s2µ(s)(u, v), u, v ∈ H1(R2). (12)

As most of the results of this work are derived for (11), let us discuss the reasons for this choice and its place in the analysis of the stability of PMLs. To our knowledge, so far only the energy estimates for the classical PMLs for non-dispersive systems are available. These results are of either of two types:

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(1) the energy estimates are derived for a PML system obtained from the change of variables (11), which, strictly speaking, is not what is used in practice. Moreover, the analysis is available only in 2D, for ψ(s) = 1 and non-dispersive models. Typically, such energy estimates result in the bound of the form kV(t)k1 ≤ C(t)kV(0)k2, with C(t) depending polynomially on t. They constitute one of the steps in

the analysis of the stability of the PML models with σ 6= const. See [8, 11, 17] for this kind of analysis. (2) the energy estimates are obtained for a PML system where the change of variables (9) is performed (again with ψ(s) = 1 and in non-dispersive case). This is indeed closer to the case of practical interest. For 1D problems these estimates are optimal and imply stability, see [18]. However, in higher dimensions they result in the bounds of the form kV(t)k1≤ C(t)kV(0)k2, with C(t) depending on t exponentially.

Thus, such estimates are non-optimal, since they do not reflect the stability of the underlying PMLs (the latter fact suggested by numerous experiments). This kind of results was obtained e.g. in [8, 19]. Our goal is to extend the results of the first type to 2D dispersive models, and show the stability of the PMLs for 1D dispersive problems for σx6= const (see Section 4). In this section we concentrate on the case σx= const.

2.2.1. Construction of Stable PMLs for Lorentz Materials in One Direction

The difficulty in the construction of the stable PML for dispersive models is the choice of the function ψx(s)

in (9). This question had been studied in [5], and for corners in particular in [20]. Let us provide several main results on the stability of PMLs, which are formulated as passivity of the corresponding sesquilinear forms, see (4). One of the choices of ψx(s) resulting in stable PMLs is offered by the following statement.

Theorem 2.4 ( [5]). Let εx, εy, µ be passive and satisfy Assumption 2.1. Then the sesquilinear form (12)

with ψx(s) = εy(s)−1 is passive for any σx≥ 0.

Remark 2.5. The passivity of the sesquilinear form (12), with ψ(s) = εy(s)−1, implies its coercivity for all

s ∈ C+. To see this, one tests (12) with su, to obtain

Re (A(u, su)) = Re  ¯ y(s)−1 (1 + s−1ε y(s)−1σx)2 k∂xuk2+ ¯sεx(s)−1k∂yuk2+ sµ(s)|s|2kuk2  = Re ¯sεy(s) −1(1 + ¯s−1ε¯ y(s)−1σx)2 |1 + s−1ε y(s)−1σx|4 k∂xuk2+ ¯sεx(s)−1k∂yuk2+ sµ(s)|s|2kuk2  = |1 + s−1εy(s)−1σx|−4 Re(¯sεy(s)−1) + 2σx|εy(s)|−2+ σ2x|εy(s)|−2Re(¯s−1ε¯y(s)−1) k∂xuk2 + Re(¯sεx(s)−1)k∂yuk2+ Re(sµ(s))kuk2.

Notice that Re(¯sε−1y ) = Re sεy−1 = Re sεy|εy|−2 > 0, for all s ∈ C+, thanks to passivity. For the same

reason Re((sεy)−1) > 0, Re(¯sε−1x ) and Re(sµ(s)) > 0, s ∈ C+. From this the coercivity of Re A(u, sv) follows.

The coercivity alone is not sufficient for the well-posedness or stability of the corresponding time-domain system. In particular, one should show the analyticity of the Laplace-domain solution in C+, see [5].

This is indeed not the only stable PML change of variables. While the theorem below does not show how to construct ψx(s), in [5] it is shown that its conditions can be ensured by choosing ψx(s) that verifies certain sign

conditions on the imaginary axis. We omit these details, since they are not important for the energy estimates. Theorem 2.6 ( [5]). Let εx, εy, µ be passive and satisfy Assumption 2.1. Let additionally

(1) ψx(s)−1 satisfy Assumption 2.1 and be passive;

(2) εx(s)εy(s)−1ψx(s)−1 be passive;

(3) µ(s)εy(s)ψx(s) be passive.

Then the sesquilinear form ˜

A(u, v) = ψx(s)εy(s)A(u, v), u, v ∈ H1(R2),

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2.2.2. Construction of Stable PMLs for Lorentz Materials in Corners

Similar results can be formulated for the corner PMLs, which are obtained by the following change of variables: x → x 1 + σxψx(s)s−1 , y → y 1 + σyψy(s)s−1 , σx, σy ≥ 0. (13)

The associated sesquilinear form reads

Ac(u, v) = εy(s)−1 (1 + s−1ψ x(s)σx)2 (∂xu, ∂xv) + εx(s)−1 (1 + s−1ψ y(s)σy)2 (∂yu, ∂yv) + s2µ(s)(u, v), u, v ∈ H1(R2). (14)

The corner PML analogue of Theorem 2.4 can be obtained by a change of variables with ψx= ε−1y and ψy= ε−1x .

Theorem 2.7 ( [5]). Let εx, εy, µ be passive and satisfy Assumption 2.1. Then the sesquilinear form Ac(u, v)

of (14) with ψx(s) = εy(s)−1 and ψy(s) = εx(s)−1 is passive for any σx≥ 0 and σy ≥ 0.

Notice that the above result covers the stability of the PML in one direction (by setting σx= 0 or σy= 0)

and in corners. A trivial analogue of Theorem 2.6 can be formulated as follows.

Theorem 2.8 ( [20]). Let εx, εy, µ be passive and satisfy Assumption 2.1. Let additionally

(1) ψx(s)−1, ψy(s)−1 be passive and satisfy Assumption 2.1;

(2) ψxεy = ψyεx;

(3) µ(s)εy(s)ψx(s) be passive.

Then the sesquilinear form ˜

Ac(u, v) = ψx(s)εy(s)Ac(u, v) = ψy(s)εx(s)Ac(u, v), u, v ∈ H1(R2),

with Ac(u, v) given by (14), is passive for any σx≥ 0 and σy≥ 0.

2.3. The Main Idea of the Energy Derivation for a PML System

There exists many ways to write a PML system for a given problem, see e.g. [8] and references therein, or [4]. To derive the energy conservation in an easy manner, it is crucial to introduce a very special set of auxiliary unknowns. One of such sets can be obtained by studying the coercivity of the corresponding sesquilinear form. 2.3.1. A Toy Example: a 2D Isotropic Non-dispersive Wave Equation (Second-Order Formulation)

Let us explain our idea based on the example of the isotropic non-dispersive 2D wave equation, to which the classical PMLs are applied in the x-direction. In the Laplace domain it reads (with ˆf being its source term):

− 1 + s−1σx −1 ∂x  1 + s−1σx −1 ∂xHˆz  − ∂2 yHˆz+ s2Hˆz= ˆf , s ∈ C+, (15)

where we assumed Hz|t=0= 0, ∂tHz|t=0= 0. The corresponding variational formulation reads

Aσx( ˆHz, v) = 1 + s

−1σ x

−2

(∂xHˆz, ∂xv) + (∂yHˆz, ∂yv) + s2( ˆHz, v) = ( ˆf , v), Hˆz, v ∈ H1(R2).

The above sesquilinear form coincides with (12) for a particular case εx = εy = µ = 1, where the change of

variables of Theorem 2.4 is performed. To show the coercivity of the sesquilinear form for all s ∈ C+, one tests

the above with v = s ˆHz (and then takes the real part), see Remark 2.5,

Aσx( ˆHz, s ˆHz) = Re  1 + s−1σx −2 (∂xHˆz, s∂xHˆz) + (∂yHˆz, s∂yHˆz) + s(s ˆHz, s ˆHz)  = Re ˆf , s ˆHz  . (16)

Testing with v = s ˆHz corresponds in the time domain to testing the corresponding equation with ∂tHz (if

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the PML. On the other hand, it is possible to obtain some energy identities directly from (16), using the Plancherel’s equality. To proceed with this idea, let us rewrite the first term of (16) in a more convenient form:

 1 + s−1σx −2 ∂xHˆz, s∂xHˆz  = ∂x ˆ Hz (1 + s−1σ x) 2, s 1 + s−1σx 2 (1 + s−1σ x) 2 ∂xHˆz ! (17) = ∂x ˆ Hz (1 + s−1σ x) 2, s + 2σx+ σ2xs−1 (1 + s−1σ x) 2 ∂xHˆz ! = (¯s + 2σx) ∂xHˆz (1 + s−1σ x)2 2 + σ2xs ∂xHˆz s (1 + s−1σ x)2 2 . (18)

Thus, (16) can be rewritten as follows:

Re (s + 2σx) ∂xHˆz (1 + s−1σ x)2 2 + σx2Re s ∂xHˆz s (1 + s−1σ x)2 2 + Re sk∂yHˆzk2+ Re sks ˆHzk2= Re ˆf , s ˆHz  . (19)

Recall the Plancherel’s identity (here η > 0) and its implication:

+∞ Z 0 e−2ηt(Hz(t), v(t))dt = 1 2πi Z η+iR ( ˆHz(s), ˆv(s))ds =⇒ +∞ Z 0 e−2ηtRe(Hz(t), v(t))dt = 1 2πi Z η+iR Re( ˆHz(s), ˆv(s))ds.

Before applying the last identity of the above to (19), let us introduce two auxiliary unknowns:

ˆ = ∂x ˆ Hz (1 + s−1σ x) 2, J =ˆ ∂xHˆz s (1 + s−1σ x) 2. (20)

This allows us to obtain a formal energy identity (assuming zero initial conditions for j, J ):

+∞ Z 0 e−2ηt (η + 2σx)kjk2+ σ2xηkJ k 2+ ηk∂ yHzk2+ ηk∂tHzk2 dt = +∞ Z 0 e−2ηt(f, ∂tHz)dt.

Importantly, this time-domain equality involves newly introduced unknowns (20). This suggests that when writing the PML system corresponding to (15), it may be advantageous to introduce the unknowns (20). Let us rewrite (15) in time using this idea (now taking f = 0 but assuming non-zero initial conditions):

∂t2Hz− ∂xj − ∂y2Hz= 0, (21a)

∂tj + 2σxj + σ2xJ = ∂x∂tHz, (21b)

∂tJ = j. (21c)

Testing (21a) with ∂tHz, we obtain

1 2 d dt k∂tHzk 2+ k∂ yHzk2 + (j, ∂x∂tHz) = 0, (j, ∂x∂tHz) (21b) = (j, ∂tj + 2σxj + σx2J ) (21c) = 1 2 d dtkjk 2+ 2σ xkjk2+ σ2x(∂tJ, J ) =1 2 d dt kjk 2+ kσ xJ k2 + 2σxkjk2.

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Combining the above, we get the following energy decay law d dtE = −2σxkjk 2, E = 1 2 k∂tHzk 2+ k∂ yHzk2+ kjk2+ σx2kJ k 2 . (22)

Indeed, when σx6= const, the formulation (21) is not perfectly matched. However, for σx= const, it is equivalent

to another, perfectly matched formulation.

2.3.2. Remaking the Toy Example: A First-Order Formulation

Our goal is to derive the PML system in the first order formulation, like the original system (5a-5g). On one hand, we would like to introduce unknowns that would correspond to s−1ˆ, s−1J (the energy of the first-orderˆ formulation for the wave equation coincides with the energy of the second-order formulation with u replaced by ∂t−1u). On the other hand, we would like to simplify the corresponding PML system of equations. First,

the system of equations in the Laplace domain with the PML change of variables, and then comment on the introduction of the time-domain unknowns:

s ˆHz= − ∂xEˆy 1 + s−1σ x + ∂yEˆx, (23a) s ˆEy = − ∂xHˆz 1 + s−1σ x , (23b) s ˆEx= ∂yHˆz. (23c)

In this first-order formulation we suggest to introduce an auxiliary unknown Ey∗(it corresponds to −s−1J ), s.t.ˆ

Ey∗= − ∂x ˆ Hz (s + σx)2 (23b) = ˆ Ey s + σx . (24)

This choice is advantageous for two reasons:

• there is no need to introduce an unknown corresponding to s−1ˆ, since it can be represented as a linear

combination of the existing unknowns

s−1ˆ = ∂x ˆ Hz (s + σx)(1 + s−1σx) (23b) = − ˆ Ey 1 + s−1σ x = −  1 − σx s + σx  ˆ Ey (24) = − ˆEy+ σxEˆy∗.

• the above linear combination can replace the first term in the right-hand side of (23a), namely 1 + s−1σx

−1

∂xEˆy

by ∂x( ˆEy− σxEˆy∗). Here we make use of the assumption σx= const.

Therefore, (23a-23c) can be rewritten in the time domain as follows:

∂tHz= −∂xEy+ σx∂xEy∗+ ∂yEx, (25a)

∂tEy+ σxEy = −∂xHz, (25b)

∂tEx= ∂yHz, (25c)

∂tE∗y+ σxEy∗= Ey. (25d)

Let us show that the energy is consistent with (22). We test the first equation above with Hz, to obtain

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The second term in the above (∂xEy− σx∂xE∗y, Hz) = −(Ey− σxEy∗, ∂xHz) (25b) = (Ey− σxEy∗, ∂tEy+ σxEy) = (Ey− σxEy∗, ∂t(Ey− σxEy∗) + σx∂tEy∗+ σxEy) (25d) = 1 2 d dtkEy− σxE ∗ yk 2+ (E y− σxEy∗, σx∂tEy∗+ σx(∂tEy∗+ σxEy∗)) = (Ey− σxEy∗, 2σx∂tE∗y) + (Ey− σxEy∗, σ2xEy∗).

Applying (25d) to each of the two terms above yields

(Ey− σxEy∗, 2σx∂tEy∗) = 2σxkEy− σxEy∗k 2, (E y− σxEy∗, σ 2 xE ∗ y) = (∂tEy∗, σ 2 xE ∗ y) = σx2 2 d dtkE ∗ yk 2.

Finally, combining (26) and the latter expressions, we obtain: d dtE = −2σxkEy− σxE ∗ yk 2, E = kHzk2+ kExk2+ kEy− σxEy∗k 2+ kσ xEy∗k 2.

As expected, the above corresponds to (22), with all the unknowns substituted by their primitives in time. 2.3.3. Remarks on the Connection Between the Formulation (25a-25d) and the B´erenger’s Split Formulation

Recall the B´erenger’s split formulation for the 2D Maxwell’s equations:

Hz = Hzx+ Hzy, (27a)

∂tEy+ σxEy= −∂xHz, (27b)

∂tEx= ∂yEx, (27c)

∂tHzx+ σxHzx= −∂xEy, (27d)

∂tHzy= ∂yEx. (27e)

To see how from (25a-25d) one can obtain the above system, we apply ∂x to (25d), and use σx= const,

∂t∂xEy∗+ σx∂xEy∗= ∂xEy.

Then we rewrite (25a) as

∂tHz= −∂xEy+ σx∂xEy∗+ ∂yEx= −∂t∂xEy∗+ ∂yEx.

Finally, we can introduce the additional unknown Hzy as in (27e). Thus, (25a-25d) can be rewritten as

∂tHz= −∂t∂xEy∗+ ∂tHzy,

∂tEy+ σxEy = −∂xHz,

∂tEx= ∂yHz,

∂t∂xEy∗+ σx∂xEy∗= ∂xEy,

∂tHzy = ∂yEx.

Then, choosing the initial conditions ∂xEy∗

t=0= − Hzx|t=0, and Hz|t=0= Hzx|t=0+ Hzy|t=0, we can ensure

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2.4. Stability of the PML of Theorem 2.7, with ψ

x

= ε

−1y

, ψ

y

= ε

−1x

In this section we extend the ideas of the construction of the PML systems of Section 2.3 to the PMLs for problems with the Lorentz dispersion (3).

2.4.1. Construction of a PML System Corresponding to Theorem 2.7, with ψx= ε−1y , ψy = ε−1x

Let us write a PML system corresponding to (5a-5g) with the PML change of variables of Theorem 2.7. We perform the PML change of variables in both directions, however, assume that σx, σy ≥ 0. Setting in the

resulting system σy = 0 and discarding the auxiliary unknowns associated with the PML in the direction y, we

will get the PML system in the direction x (similarly we can obtain the PML system in the direction y). Let us start with the first equation (5a), which in the Laplace domain with the PML change of variables of Theorem 2.7 and with the use of the constitutive relation s ˆDx= sεxEˆx reads

s ˆDx= sεxEˆx=  1 + σy sεx −1 ∂yHˆz.

In the time-domain this is equivalent to ∂tDx+ σyEx= ∂yHz.

Remark 2.9. When deriving the PML system, one ignores all the source terms and initial conditions, since they are necessarily supported outside of the domain where the PML is used.

The equation (5b) can be treated similarly. To deal with (5c), we suggest to employ the splitting idea of B´erenger [1]. This is motivated by results of Section 2.3.3. Let us introduce split fields ˆHzx and ˆHzy so that

they satisfy in the Laplace domain the following equations:

sεx(s) ˆHzy= ∂yEˆx, sεy(s) ˆHzx= −∂xEˆy, s ˆBz= ∂yEˆx− ∂xEˆy = sεx(s) ˆHzy+ sεy(s) ˆHzx. (28)

Applying the PML change of variables of Theorem 2.7 to the above equations, we obtain the following: sεxHˆzy+ σyHˆzy= ∂yEˆx, sεyHˆzx+ σxHˆzx= −∂xEˆy,

s ˆBz= sεx(s) ˆHzy+ sεy(s) ˆHzx= ∂yEˆx− ∂xEˆy− σyHˆzy− σxHˆzx.

Finally, the time-domain expression of the operators sεx(s), sεy(s) is done similarly to the system (5a-5g), via

the introduction of the auxiliary unknowns. After the PML change of variables, (5a-5g) becomes:

∂tDx+ σyEx= ∂yHz, (29a) ∂tDy+ σxEy = −∂xHz, (29b) ∂tBz+ σyHzy+ σxHzx= ∂yEx− ∂xEy, (29c) ∂tHzx+ ny X `=0 εy`hx`+ σxHzx= −∂xEy, (29d) ∂tHzy+ nx X `=0 εx`hy`+ σyHzy = ∂yEx, (29e) ∂thα`+ ωβ`2 bα`= Hzα, ∂tbα`= hα`, ` = 0, . . . , nβ, (α, β) ∈ {(x, y), (y, x)}, (29f)

which we equip with the equations (5d − 5g). In the derivation of the above system we did not use the assumption σx, σy = const, hence it is perfectly matched. However, even in the case σx, σy = const, obtaining

the energy estimates for it directly requires to solve additional difficulties, besides the dispersive behaviour. We will report how to resolve them elsewhere. To simplify the discussion, we suggest to consider another formulation, equivalent to (29a-29f) for constant absorption parameters (see Remark 2.10 or Section 2.3.3). It

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differs from (29a-29f) only by treatment of the equation for Bz (5c). More precisely, we introduce ˆEx∗, ˆEy∗, s.t.

ˆ

Hzy= ∂yEˆx∗, ˆHzx= −∂xEˆ∗y, cf. (28). Then, formally, using σx= const, σy= const,

sεxEˆ∗x+ σyEˆx∗= ˆEx, sεyEˆy∗+ σxEˆy∗= ˆEy, s ˆBz= ∂yEˆx− ∂xEˆy− σy∂yEˆx∗+ σx∂xEˆy∗. (30)

The realization of the operators sεx, sεyis done similarly to (5a-5g). We thus obtain the following PML system:

∂tDx+ σyEx= ∂yHz, (31a) ∂tDy+ σxEy= −∂xHz, (31b) ∂tBz+ σy∂yEx∗− σx∂xEy∗= ∂yEx− ∂xEy, (31c) ∂tEx∗+ nx X `=0 εx`λ∗x`+ σyEx∗= Ex, (31d) ∂tEy∗+ ny X `=0 εy`λ∗y`+ σxEy∗= Ey, (31e) ∂tλ∗α`+ ω 2 α`p∗α`= Eα∗, ∂tp∗α`= λ∗α`, ` = 0, . . . , nα, α ∈ {x, y}, (31f) ∂tDα= ∂tEα+ nα X `=0 εα`λα`, (31g) ∂tλα`+ ω2α`pα`= Eα, ∂tpα`= λα`, ` = 0, . . . , nα, α ∈ {x, y}, (31h) ∂tBz= ∂tHz+ nµ X `=0 µ`λµ`, (31i) ∂tλµ`+ ωµ`2 pµ`= Hz, ∂tpµ`= λµ`, ` = 0, . . . , nµ. (31j)

In the above we introduced new PML unknowns E∗

x, (λ∗x`, p∗x`), ` = 0, . . . , nxand Ey∗, (λ∗y`, p∗y`), ` = 0, . . . , ny.

Remark 2.10. The equivalence of (29a-29f) and (31a-31j) for constant σx, σy, i.e. that with properly chosen

initial data, the solution E, D, Bz, Hz of (29a-29f) solves (31a-31j) and vice versa, can be shown as in [8].

Remark 2.11. In the system (31a-31j) the PML in the direction y is realized via the introduction of the auxiliary unknowns Ey∗, λ∗y`, p∗y`, ` = 0, . . . , ny. To obtain the PML in the direction x only, one sets σy= 0. In

this case (31e) and (31f) for α = y are decoupled from the rest of the equations and can be discarded. 2.4.2. Stability of the PML System (31a-31j)

The energy non-growth result is summarized in the following theorem. Theorem 2.12. An energy associated with (31a-31j) defined as

E = 1 2 kEx− σyE ∗ xk 2+ kσ yE∗xk 2+ kE y− σxEy∗k 2+ kσ xE∗yk 2+ kH zk2 + Ex∗+ Ey∗+ Ez, (32) E∗ x= 1 2 nx X `=0 εx` kλx`− σyλ∗x`k 2+ ω2 x`kpx`− σyp∗x`k 2+ kσ yλ∗x`k 2+ ω2 x`kσyp∗x`k 2 , Ey∗= 1 2 ny X `=0 εy` kλy`− σxλ∗y`k 2+ ω2 y`kpy`− σxp∗y`k 2+ kσ xλ∗y`k 2+ ω2 y`kσxp∗y`k 2 , Ez= 1 2 nµ X `=0 µ` kλµ`k2+ ωµ`2 kpµ`k2 ,

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does not grow, provided σx, σy≥ 0: d dtE = −2σykEx− σyE ∗ xk 2− 2σ xkEy− σxEy∗k 2.

Proof. First, we test (31c) with Hz to obtain

(∂tBz, Hz) − (∂yEx− σy∂yEx∗, Hz) + (∂xEy− σx∂xE∗y, Hz) = 0. (33)

Let us consider each term of the above expression separately. The first term is exactly as in (7):

(∂tBz, Hz) (7) = d dt  1 2kHzk 2+ E z  .

The second term, after the integration by parts, becomes −(∂yEx− σy∂yE∗x, Hz) = (Ex− σyEx∗, ∂yHz) (31a) = (Ex− σyEx∗, ∂tDx+ σyEx) = (Ex− σyEx∗, ∂tDx) + σy(Ex− σyEx∗, Ex− σyEx∗+ σyEx∗) = (Ex− σyEx∗, ∂tDx) + σykEx− σyE∗xk 2+ σ2 y(Ex− σyEx∗, E ∗ x) = T1+ σykEx− σyEx∗k 2+ T 2. (34)

Let us first consider

T1= (Ex− σyEx∗, ∂tDx) (31g) = Ex− σyEx∗, ∂tEx+ nx X `=0 εx`λx` ! = (Ex− σyEx∗, ∂t(Ex− σyEx∗) + σy∂tEx∗) + Ex− σyEx∗, nx X `=0 εx`λx` ! (31d) = 1 2 d dtkEx− σyE ∗ xk2+ σy Ex− σyE∗x, Ex− σyE∗x− nx X `=0 εx`λ∗x` ! + Ex− σyEx∗, nx X `=0 εx`λx` ! = 1 2 d dtkEx− σyE ∗ xk 2+ σ ykEx− σyEx∗k 2+ nx X `=0 εx`(Ex− σyEx∗, λx`− σyλ∗x`) .

The last term in the above corresponds to a weighted sum of norms of a linear combination of auxiliary unknowns:

nx X `=0 εx`(Ex− σyEx∗, λx`− σyλ∗x`) (31h,31f ) = nx X `=0 εx` ∂t(λx`− σyλ∗x`) + ωx`2 (px`− σyp∗x`), λx`− σyλ∗x`  (31h,31f ) = 1 2 d dt nx X `=0 εx`kλx`− σyλ∗x`k 2+ nx X `=0 εx`ω2x`(px`− σypx`∗ , ∂t(px`− σyp∗x`)) =1 2 d dt nx X `=0 εx` kλx`− σyλ∗x`k 2+ ω2 x`kpx`− σyp∗x`k 2 . (35)

Summarizing the above,

T1= 1 2 d dtkEx− σyE ∗ xk 2+ σ ykEx− σyEx∗k 2+1 2 d dt nx X `=0 εx` kλx`− σyλ∗x`k 2+ ω2 x`kpx`− σyp∗x`k 2 .

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It remains to obtain an explicit energy-like expression for the term T2 in (34): T2= σy2(Ex− σyEx∗, E∗x) (31d) = σy2 ∂tEx∗+ nx X `=0 εx`λ∗x`, E∗x ! (31f ) = σ 2 y 2 d dt kE ∗ xk2+ nx X `=0 εx` kλ∗x`k2+ ωx`2 kp∗x`k2  ! ,

where the derivation of the latter identity is done similarly as e.g. in (7) or (35). Summarizing the above, we obtain the following expression for the second term in (33):

−(∂yEx− σy∂yEx∗, Hz) = 1 2 d dt kEx− σyE ∗ xk 2+ kσ yEx∗k 2 + 2σ ykEx− σyEx∗k 2+ d dtE ∗ x.

Repeating almost verbatim the above computations, one can show that the third term in (33) reads:

(∂xEy− σx∂xEy∗, Hz) = 1 2 d dt kEy− σxE ∗ yk2+ kσxE∗yk2 + 2σxkEy− σxEy∗k2+ d dtE ∗ y.

Summing up all the terms of (33) we obtain the desired statement.  Remark 2.13. The above result can be generalized to the case of Lorentz materials with dissipation, i.e.

εα(s) = 1 + nα X `=0 εα` s2+ 2ν α`s + ωα`2 , εα`> 0, να`≥ 0, ωα`≥ 0, ` = 0, . . . , nα, α ∈ {x, y}, µ(s) = 1 + nµ X `=0 µ` s2+ 2ν µ`s + ω2µ` , µ`> 0, νµ`≥ 0, ωµ`≥ 0, ` = 0, . . . , nµ.

In this case the choice of ψx(s) = εy(s)−1, ψy = εx(s)−1 would lead to a stable system.

Remark 2.14. With the help of the Young’s inequality, one can show that the norm of the field E is controlled with the help of the stability result of Theorem 2.12. E.g. consider the first term in (32):

kEx− σyEx∗k 2+ kσ yEx∗k 2≥ kE xk2+ 2kσyEx∗k 2− 2kE xkkσyE∗xk ≥ kExk2+ 2kσyEx∗k 22 3kExk − 3 2kσyE ∗ xk = 1 3kExk 2+1 2kσyE ∗ xk 2.

2.5. Stability of the PML of Theorem 2.8, for General ψ

x

(s), ψ

y

(s)

An exact expression of the corresponding to the PML with the change of variables as in Theorem 2.8 is slightly more complicated. It is easier to express the energy of this formulation in terms of new, ’effective’ electric and magnetic fields, rather than the original unknowns. However, this requires the reformulation of the corresponding system without the PML. Similar ideas are used in [14].

2.5.1. A Preliminary Reformulation of the System (5a-5g)

Let us rewrite the Maxwell’s equations in a more convenient form, equivalent to the original Lorentz system (5a-5g), which, however, would allow to handle easier the energy of the corresponding PML system. In particular, notice that the PML change of variables of Theorem 2.8 results in a passivity of a special scaled sesquilinear form. This form can be viewed as a sesquilinear form resulting from the PML of Theorem 2.7 applied to the problem with (passive) coefficients εe

y = ψx−1, εex= εxψ−1x ε−1y = ψ−1y and µe= εyψxµ.

Let us consider the solution (E, Hz) of the system (5a-5g) with the initial conditions (E, Hz)|t=0= (E0, Hz0).

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rest of unknowns. Then, in the Laplace domain, the system (5a-5g) reads

sεx(s) ˆEx− Ex0= ∂yHˆz, sεy(s) ˆEy− Ey0= −∂xHˆz, sµ(s) ˆHz− Hz0= ∂yEˆx− ∂xEˆy.

The above can be split into two systems:

sεx(s) ˆEx(1)= ∂yHˆz(1), sεy(s) ˆEy(1)= −∂xHˆz(1), sµ(s) ˆH (1) z − Hz0= ∂yEˆx(1)− ∂xEˆy(1), (36) and sεx(s) ˆEx(2)− Ex0= ∂yHˆz(2), sεy(s) ˆEy(2)− Ey0= −∂xHˆz(2), sµ(s) ˆH (2) z = ∂yEˆx(2)− ∂xEˆ(2)y . (37) Due to linearity, E = E(1) + E(2), H z = H (1)

z + Hz(2). Let us consider the first equation of (36), which we

multiply by ψx(s)−1εy(s)−1, to obtain

sεx(s)ψx(s)−1εy(s)−1Eˆx(1)= ∂yψx−1(s)εy(s)−1Hˆz(1)= ∂yHˆze, Hˆ e

z= ψx−1(s)εy(s)−1Hˆz(1).

Here the index ’e’ stands for ’effective’. Setting εe

x(s) = εx(s)ψx(s)−1εy(s)−1, we rewrite the above as

sεex(s) ˆE (1)

x = ∂yHˆze. (38)

Next, we repeat the procedure with the second equation of (36), defining εe

y = ψx(s)−1:

sεey(s) ˆEy(1)= −∂xHˆze. (39)

Finally, for the third equation of (36), it suffices to introduce µe(s) = ψ

x(s)εy(s)µ(s) and rewrite

sµ(s)ψx(s)εy(s)ψx(s)−1εy(s)−1Hˆz(1)− Hz0= sµe(s) ˆHze− Hz0= ∂yEˆx(1)− ∂xEˆy(1). (40)

Therefore, the system (36) is equivalent to the system of equations (38, 39, 40), provided that the initial conditions are chosen as He

z(0) = Hz0. Similarly the system (37) can be rewritten:

sεex(s) ˆEex− Ex0= ∂yHˆz(2), sεeyEˆye− Ey0= −∂xHˆz(2), sµe(s) ˆHz(2)= ∂yEˆxe− ∂xEˆye, with ˆEe x = εy(s)ψx(s) ˆE (2) x , ˆEye = εy(s)ψx(s) ˆE (2)

y . Since the above two systems have similar structures, from

now on we will concentrated only on one of them, namely (38, 39, 40). Thanks to Theorem 2.8, the quantities εe

x, εey, µe are passive. Therefore, they are of the following form, see Theorem 2.2:

εeα(s) = 1 + ne α X `=0 εe α` s2+ (ωe α`) 2, ε e α`> 0, ω e α`∈ R, ` = 0, . . . , n e α, α ∈ {x, y}, µ(s) = 1 + ne µ X `=0 µ` s2+ (ωe µ`)2 , µe`> 0, ωµ`e ∈ R, ` = 0, . . . , neµ.

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The system without the PML mimics the Lorentz system (5a-5g): ∂tDxe= ∂yHze, ∂tDey= −∂xHze, ∂tBze= ∂yE(1)x − ∂xEy(1), ∂tDαe = ∂tEα(1)+ neα X `=0 εeα`λeα`, ∂tλeα`+ (ωeα`)2peα`= Eα(1), ∂tpeα`= λeα`, ` = 0, . . . neα, α ∈ {x, y}, ∂tBze= ∂tHze+ ne µ X `=0 µe`λeµ`, ∂tλeµ`+ (ω e µ`) 2pe µ`= H e z, ∂tpeµ` = λ e µ`, ` = 0, . . . n e µ. (41)

Then the following result holds true.

Proposition 2.15. Let (E(1), Hze, Bze) solve the system (41) with the initial conditions chosen as Hze|t=0=

Bze|t=0= Hz0 and as zero for the rest of unknowns. Let E, Hz be the solution to the system (5a-5g) with the

initial conditions chosen as Hz|t=0= Bz|t=0= Hz0 and as zero for the rest of unknowns. Then E(1)(t) = E(t)

and Bez(t) = Bz(t) for all t ≥ 0.

To recover Hz(1), we can use the identity Bz= Bze, or sµ(s) ˆH (1)

z = sµe(s) ˆHze, which gives in the time domain:

∂tHz(1)+ nµ X `=0 µ`λ (1) µ` = ∂tHze+ nµ X `=0 µ`λeµ`, (42) ∂tλ (1) µ` + ω 2 µ`p (1) µ` = H (1) z , ∂tp (1) µ` = λ (1) µ`, ` = 0, . . . nµ. (43)

With the system (41) we can associate a conservation of a certain energy. Theorem 2.16. An energy E of (41), defined below, satisfies dtdE = 0. Here

E = 1 2  kEx(1)k 2 + kEy(1)k 2 + kHzek 2 + Ex+ Ey+ Ez, Eα= 1 2 nα X `=0 εeα` kλe α`k 2+ (ωe α`) 2kpe α`k 2 , α ∈ {x, y}, E z= 1 2 nµ X `=0 µe` kλe µ`k 2+ (ωe µ`) 2kpe µ`k 2 .

As for the field Hz(1), we were not able to deduce the conservation of its norm from the equations (42-43).

However, the following stability result which will be of use later.

Proposition 2.17. The following bound holds for the solution of the system (42-43) coupled with (41):

˜ Ez(t) := 1 2 kH (1) z (t) − H e z(t)k 2+ nµ X `=0 µ`  kλ(1)µ`(t)k2+ ω2µ`kp(1)µ`(t)k2 ! ≤ 2 ˜Ez(0) + CE (0)t2, t ≥ 0,

where E (t) is defined in Theorem 2.16 and C > 0 is constant.

Proof. See Appendix A. 

Remark 2.18. The result of (at most) linear growth of the field Hz(1) is non-optimal, at least when the initial

conditions are chosen as in Proposition 2.15. This follows from the fact that Hz(1) solves the original Maxwell

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2.5.2. The PML System and Its Stability

The application of the PML of Theorem 2.8 to (41) results in the system that has the same structure as (31a-31j). Indeed, ψx= (εey)−1 and ψy= (εex)−1. We can immediately write the corresponding PML system:

∂tDxe+ σyEx(1)= ∂yHze, ∂tDey+ σxEy(1)= −∂xHze, ∂tBze= ∂yEx(1)− σy∂yEx∗− ∂xEy(1)+ σx∂xE∗y, ∂tEx∗+ ne x X `=0 εex`λ∗x`+ σyEx∗= Ex(1), ∂tEy∗+ ney X `=0 εey`λ∗y`+ σxEy∗= Ey(1), ∂tλ∗α`+ (ω e α`) 2pe α`= E (1) α , ∂tp∗α`= λ ∗ α`, ` = 0, . . . , nα, α ∈ {x, y}, ∂tDαe = ∂tEα+ ne α X `=0 εeα`λeα`, ∂tλeα`+ (ω e α`) 2pe α`= E (1) α , ∂tpeα`= λ e α`, ` = 0, . . . n e α, α ∈ {x, y}, ∂tBze= ∂tHze+ neµ X `=0 µe`λeµ`, ∂tλeµ`+ (ωeµ`)2peµ`= Hze, ∂tpeµ`= λeµ`, ` = 0, . . . neµ. (44)

As before, the physical field Hz(1) can be recovered using (42-43). An energy associated with the above system

can be derived as in Theorem 2.12, and the bound on Hz(1) as in Proposition 2.17.

Theorem 2.19. An energy associated with (44) defined as

E = 1 2  kEx(1)− σyEx∗k 2+ kσ yEx∗k 2+ kE(1) y − σxEy∗k 2+ kσ xEy∗k 2+ kHe zk 2+ Ee,∗ x + E e,∗ y + E e z, (45) Exe,∗ = 1 2 nx X `=0 εex` kλ e x`− σyλ∗x`k 2 + (ωx`e ) 2 kpex`− σyp∗x`k 2 + kσyλ∗x`k 2 + (ωx`e ) 2 kσyp∗x`k 2 , Ee,∗ y = 1 2 ny X `=0 εey` kλe y`− σxλ∗y`k 2+ (ωe y`) 2kpe y`− σxp∗y`k 2+ kσ xλ∗y`k 2+ (ωe y`) 2 xp∗y`k 2 , Ee z = 1 2 nµ X `=0 µe` kλe µ`k 2+ (ωe µ`) 2kpe µ`k 2

does not grow, provided σx, σy≥ 0:

d dtE = −2σykE (1) x − σyEx∗k 2− 2σ xkEy(1)− σxEy∗k 2.

Moreover, for the solution of (42-43) coupled with (44), it holds

˜ Ez(t) := 1 2 kH (1) z (t) − H e z(t)k 2+ nµ X `=0 µ`  kλ(1)µ`(t)k2+ ω2µ`kp(1)µ`(t)k2 ! ≤ 2 ˜Ez(0) + CE (0)t2, t ≥ 0, where C > 0 is constant.

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3. General Passive Materials: Stability with Energy Techniques

In this section we summarize some results on the materials whose dielectric permittivity and magnetic permeability do not satisfy the conditions of Section 2. One of the examples is given by Warburg’s law, when the dielectric permittivity ε(s) = 1 + 1+a

√ s

s(1+b√s), a, b > 0, see e.g. [21,22] for the corresponding numerical method. The construction of the PML for this types of systems is done as before, via a special change of variables (9). In this section we will show the stability of the following PML.

Theorem 3.1 ( [5]). Let εx, εy, µ be passive. Then the sesquilinear form (14) with ψx(s) = εy(s)−1 and

ψy(s) = εx(s)−1 is passive for any σx≥ 0 and σy ≥ 0.

The stability of the corresponding system without the PML was shown in [14] with the help of a Herglotz-Nevanlinna representation formula. This integral representation allows to write any passive system (even dissi-pative) in the form (5a-5g), and thus, associate with any system an energy that is conserved (while at the same time, there may exist another energy that decays). A conservative extension for a general class of dissipative systems had been constructed, though in a slightly different form, in the seminal work by Figotin and Schenker in [23]. Such an approximation of an arbitrary material with the help of Lorentz materials will be used to construct a PML system similar to (31a-31j), see Section 3.2.

3.1. Time-Domain System without PMLs

The results of this section are due to [14], where it is suggested that passive functions corresponding to the Laplace transform of real-valued in the time domain distributions (i.e., in the Laplace domain, ˆf (s) =

ˆ

f (s), s ∈ C+) satisfy the property formulated below. The following lemma is nothing more but a version of the

Herglotz-Nevanlinna representation for such functions.

Lemma 3.2 (Herglotz-Nevanlinna Representation). The function f (s), which satisfies f (s) = f (s), with s ∈ C+, is passive if and only if there exists a non-decreasing function of a bounded variation νf(ξ), for which the

Stiltjes integral

R

−∞ dνf(ξ)

1+ξ2 is finite and f (s) is represented as the following Stiltjes integral:

f (s) = a + ∞ Z −∞ dνf(ξ) s2+ ξ2, a ≥ 0. (46) Here a = lim sr→+∞

f (sr), and the measure νf satisfies +∞

R

−∞ dνf(ξ)

1+ξ2 < ∞.

Proof. Notice that f (s) is passive if and only if Re (−iωf (−iω)) > 0 for Im ω > 0, or Im(ωf (−iω)) > 0 for Im ω > 0. Hence, ωf (−iω) is Herglotz, and one can apply the results of [14, Section 4.1] to f (−iω).  For instance, to obtain the Lorentz magnetic permeability (3), we can take a sum of δ-measures νµ(ξ) = nµ

P

`=0

µ`δ(ξ − ωµ`). Without loss of generality, let us assume that [6, 14]

εx(s) → 1, εy(s) → 1, µ(s) → 1, as Re s → +∞.

Then Lemma 3.2 yields (where we take into account εx(sr), εy(sr), µ(sr) → 1 as sr→ +∞)

εα(s) = 1 + +∞ Z −∞ dνα s2+ ξ2, α ∈ {x, y}, µ(s) = 1 + +∞ Z −∞ dνµ s2+ ξ2, s ∈ C+.

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These identities generalize the corresponding definitions for the Lorentz materials (3), with the exception that the weighted sums are substituted by integrals with Borel measures. Our goal is to write the system (1) in the time domain, using the above representation of ε, µ. Let us recall from [14] how the equation s ˆDx= sεx(s) ˆEx

can be written in the time domain. Let us introduce ˆλx(ξ) and ˆpx(ξ), which map ξ ∈ R into a set of analytic

functions from s ∈ C+ into L2(R2), so that they satisfy the following identities:

sˆλx(ξ) + ξ2pˆx(ξ) = ˆEx, sˆpx(ξ) = ˆλx(ξ), ξ ∈ R.

Then, in the time domain, we can rewrite (formally) s ˆDx= sε ˆEx as:

∂tDx= ∂tEx+ ∞ Z −∞ λx(ξ)dνx(ξ), ∂tλx(ξ) + ξ2px(ξ) = Ex, ∂tpx(ξ) = λx(ξ), ξ ∈ R. (47)

If the measure νx has a density fνx, we can define the quantities λx(ξ), px(ξ) only for ξ ∈ supp fνx(ξ) (this is

the case for the Lorentz materials, where the density has a discrete support). Thus, extending the above idea to the rest of the constitutive relations, we write (1) in the form (5a-5g):

∂tDx= ∂yHz, ∂tDy= −∂xHz, ∂tBz= ∂yEx− ∂xEy, ∂tDα= ∂tEα+ ∞ Z −∞ λα(ξ)dνα(ξ), ∂tλα+ ξ2pα= Eα, ∂tpα= λα, ξ ∈ R, α ∈ {x, y}, ∂tBz= ∂tHz+ ∞ Z −∞ λµ(ξ)dνµ(ξ), ∂tλµ+ ξ2pµ= Eµ, ∂tpµ= λµ, ξ ∈ R. (48)

Here λα(ξ), pα(ξ), α ∈ {x, y, µ}, are functions from R to C R; L2(R2). An associated energy is conserved [14].

Proposition 3.3. An energy E associated with (48), defined below, satisfies dtdE = 0. Here

E = 1 2 kExk 2+ kE yk2+ kHzk2 + Ex+ Ey+ Ez, Eα= 1 2 ∞ Z −∞ kλαk2+ ξ2kpαk2 dνα(ξ), α ∈ {x, y}, Ez= 1 2 ∞ Z −∞ kλµk2+ ξ2kpµk2 dνµ(ξ).

3.2. The PML System and Its Stability

Let us perform the PML change of variables for the system (48) as per Theorem 3.1. This results in the following analogue of the system (31a-31j) (where again the difference is that the sums are substituted by

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integrals). The derivation of this system follows the derivation of (31a-31j), combined with the use of (47). ∂tDx+ σyEx= ∂yHz, ∂tDy+ σxEy= −∂xHz, ∂tBz+ σy∂yEx∗− σx∂xEy∗= ∂yEx− ∂xEy, ∂tDα= ∂tEα+ ∞ Z −∞ λα(ξ)dνα(ξ), ∂tλα+ ξ2pα= Eα, ∂tpα= λα, ξ ∈ R, α ∈ {x, y}, ∂tEx∗+ ∞ Z −∞ λ∗x(ξ)dνx(ξ) + σyEx∗= Ex, ∂tEy∗+ ∞ Z −∞ λ∗y(ξ)dνy(ξ) + σxEy∗= Ey, ∂tλ∗α+ ξ2p∗α= Eα∗, ∂tp∗α= λ∗α, ξ ∈ R, α ∈ {x, y}, ∂tBz= ∂tHz+ ∞ Z −∞ λµ(ξ)dνµ(ξ), ∂tλµ+ ξ2pµ= Eµ, ∂tpµ= λµ, ξ ∈ R. (49)

We can formulate the following analogue of Theorem 2.12, with the proof almost verbatim the same. Theorem 3.4. An energy E associated with (49), defined as

E = 1 2 kEx− σyE ∗ xk 2+ kσ yEx∗k 2+ kE y− σxEy∗k 2+ kσ xEy∗k 2+ kH zk2 + Ex+ Ey+ Ez, Ex= 1 2   +∞ Z −∞ kλx− σyλ∗xk2+ ξ2kpx− σyp∗xk2+ kσyλ∗xk2+ ξ2kσyp∗xk2 dνx  , Ey= 1 2   +∞ Z −∞ kλy− σxλ∗yk 2+ ξ2 kpy− σxp∗yk 2+ kσ xλ∗yk 2+ ξ2 kσxp∗yk 2 dν y  , Ez= 1 2 +∞ Z −∞ kλµk2+ ξ2kpµk2 dνµ,

does not grow, provided any σx, σy ≥ 0:

d

dtE = −2σxkEy− σxE

yk2− 2σykEx− σyEx∗k2.

4. Stability of PMLs for a Non-constant Absorption Parameter for the 1D

Dispersive System

While we have proven the stability of the new PMLs under the assumption σx= const, obtaining the energy

estimates for σx 6= const remains an open question. It seems that showing the stability for varying σx is not

too trivial even for 1D dispersive models (being almost obvious for classical PMLs for a 1D non-dispersive wave equation). The goal of this section is to prove the stability of the PMLs, that were proposed and analyzed in [5] and in this work for σx≡ const, in the case when σx6= const for a 1D dispersive system of equations.

All over this section, without loss of generality, we will assume that the permittivity and the permeability are Lorentz (3). Moreover, we will consider only the PMLs suggested in Theorem 2.7. However, all the results can be extended to more general εx, εy, µ, and other PMLs (e.g. given in Theorem 3.1).

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The 1D analogue of the dispersive system of equations (5a-5g) in the Laplace domain reads:

sεy(s) ˆEy = −∂xHˆz, sµ(s) ˆHz= −∂xEˆy. (50)

The time-domain equivalent of the above is given by the system of equations ∂tDy= −∂xHz, ∂tBz= −∂xEy, ∂tDy= ∂tEy+ ny X `=0 εy`λy`, ∂tλy`+ ω2y`py`= Ey, ∂tpy`= λy`, ` = 0, . . . , ny, ∂tBz= ∂tHz+ ny X `=0 µ`λµ`, ∂tλµ`+ ω2µ`pµ` = Hz, ∂tpµ`= λµ`, ` = 0, . . . , nµ. (51)

This section is organized as follows. First, we consider a simplified case εy = µ, for which the derivation of

the energy estimates can be done as for the 1D non-dispersive wave equation. Next, we will concentrate on a general case εy 6= µ, and show how to deal with it by rewriting the PML system of equations in a different form

and deriving the energy estimates for this new system.

4.1. Stability of PMLs with ψ

x

= ε

−1y

for a Special Case ε

y

= µ

Let us assume that εy(s) = µ(s), and apply the PML change of variables (9) with ψx(s) = εy(s)−1 to (50):

sεy(s) ˆEy= −  1 + σx sεy(s) −1 ∂xHˆz, sεy(s) ˆHz= −  1 + σx sεy(s) −1 ∂xEˆy.

This results in the following system:

sεy(s) ˆEy+ σxEˆy= −∂xHˆz, sεy(s) ˆHz+ σxHˆz= −∂xEˆy, x ∈ R.

In the time domain, the above reads:

∂tDy+ σxEy= −∂xHz, ∂tBz+ σxHz= −∂xEy, (52a) ∂tDy= ∂tEy+ ny X `=0 εy`λy`, ∂tλy`+ ωy`2py`= Ey, ∂tpy`= λy`, (52b) ∂tBz= ∂tHz+ ny X `=0 εy`λz`, ∂tλz`+ ωy`2 pz`= Hz, ∂tpz` = λz`, ` = 0, . . . , ny. (52c)

The energy nongrowth result follows trivially.

Proposition 4.1. An energy E associated with the system (52a-52c) defined as

E = kEyk2+ kHzk2+ Ey+ Ez, Ey= ny X `=0 εy` kλy`k2+ ωy`2 kpy`k2 , Ez= ny X `=0 εy` kλz`k2+ ω2y`kpz`k2 ,

does not grow, provided any non-negative σx∈ L∞(R):

d dtE = − +∞ Z 0 |Ey|2+ |Hz|2 σx(x)dx.

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Proof. The energy identity is deduced by testing the first equation of (52a) with Ey, and the second equation

of (52a) with Hz, and summing up the obtained identities. The details are left to the reader. 

4.2. Stability of PMLs with ψ

x

= ε

−1y

for a General Case when ε

y

6= µ

The case εy 6= µ is less trivial. Applying the PML change of variables (9) with ψx = ε−1y to (50), we can

rewrite the corresponding system in the second order formulation, cf. (10),

∂x εy(s)−1  1 + σx(x) sεy(s) −1 ∂xHˆz ! − s2µ(s)  1 + σx(x) sεy(s)  ˆ Hz= 0, x ∈ R,

with σ(x) ≡ 0 for x < 0. Following the ideas of Section 2.3, we will show the coercivity of the corresponding sesquilinear form for all s ∈ C+, namely

A( ˆHz, v) =  ε y(s)−1 1 + σx(x)(sεy(s))−1 ∂xHˆz, ∂xHˆz  + s2µ(s) 1 + σx(x)(sεy(s))−1  ˆ Hz, v  , ˆHz, v ∈ H1(R). (53)

So far, we were able to show only a very special estimate, derived by testing (53) with v = s ˆHz and using a

proper scaling. Defining the branch cut of the square root as R≤0, we multiply the above with

q

sεy(s)

sµ(s) and test

the result with s ˆHz. This results in the following:

Rer sεy sµA  ˆHz, s ˆHz = Re   ¯ sqsεy sµε −1 y  1 + σx(x)(sεy)−1  |1 + σ(x)(sεy)−1| 2 ∂xHˆz, ∂xHˆz   + Re  ¯ sr sεy sµs 2µ  1 + σx(x) sεy  ˆ Hz, ˆHz  .

We use the notation qsεy

sµ to underline that the real parts of the numerator and denominator are positive in

C+, and hence sεy

sµ does not cross the branch cut of the square root when s ∈ C+. In this case we can use the

identity√z1z2=

√ z1

z2for z1, z2∈ C+, and rewrite the above as follows:

Rer sεy sµA  ˆHz, s ˆHz = Re  |s|2((sµ)(sε y))− 1 2 ∂xHˆz 1 + σx(x)(sεy)−1 2  + Re  |εy|−2((sµ)−1(sεy)) 1 2 Z R σx(x) ˆ Hz 1 + σx(x)(sεy)−1 2 dx   + Re  |s|2((sε y)(sµ)) 1 2 ˆ Hz 2 + |s|2((sεy)−1(sµ)) 1 2 Z R σx(x) ˆ Hz 2 dx  . (54)

Thanks to Rep(sµ)(sεy) > 0 and Rep(sµ)−1(sεy) > 0 for s ∈ C+ we can see that the real part of every term

in the above is positive. This implies the coercivity of the scaled sesquilinear form A( ˆHz, v). Moreover, the

apparatus of the work [5] can be used to show the well-posedness and the stability of the resulting PML system in the time domain. However, to perform the energy analysis, it is necessary to rewrite the original system of 1D equations in a very special form. This is the subject of the following section.

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4.2.1. A Preliminary Reformulation of the System (51)

For constant σx = const, the estimate (54) would imply the passivity of the scaled sesquilinear form

q

y

sµA( ˆHz, v), cf. Theorem 2.8. This suggests that to write the time-domain PML system for which the

energy estimates are derived in an easy manner, it may be first necessary to reformulate the original system without the PML (51) in an equivalent form, making use of the ideas of Section 2.5.1.

As in Section 2.5.1, we will make use of the linearity, and rewrite the corresponding system in the time domain, assuming the initial conditions (Ey, Hz)|t=0 = (0, Hz0). The case (Ey0, 0) can be treated similarly.

We thus start with (50) with the initial conditions (0, Hz0), using the same notation as in Section 2.5.1:

sεyEˆy(1)= −∂xHˆz(1), sµ ˆH (1)

z − Hz0= −∂xEˆy,

and then define ˆHe

z =p(sεy)−1(sµ) ˆH (1) z . Then q (sµ)(sεy) ˆE(1)y = −∂xHˆze, q (sµ)(sεy) ˆHze− Hz0= −∂xEˆy(1). (55)

Since for µ 6= εy, the functionp(sµ)(sεy) is not rational, to rewrite the above system in the time domain, we

will make use of the ideas of Section 3. In particular, notice that Rep(sεy)(sµ) > 0 for s ∈ C+. Lemma 3.2

applied to s−1p(sεy)(sµ), with the use of lim sr→+∞  s−1p(sεy)(sµ)  = 1, yields s−1 q (sεy)(sµ) = 1 + +∞ Z −∞ dν(ξ) s2+ ξ2.

Then the system (55) in the time domain reads

∂tDey= −∂xHze, ∂tBze= −∂xE(1)y , ∂tDey= ∂tEy(1)+ +∞ Z −∞ λey(ξ)dν(ξ), ∂tλey+ ξ 2pe y = E (1) y , ∂tpey= λ e y, ∂tBze= ∂tHze+ +∞ Z −∞ λez(ξ)dν(ξ), ∂tλez+ ξ 2pe z= H e z, ∂tpez= λ e z. (56)

Let us remark that here we set Hze|t=0= Hz0 and Bze|t=0= Hz0. As before, to recover H (1)

z , we can make use

of the identities similar to (42-43), more precisely

∂tHz(1)+ nµ X `=0 µ`λµ`= ∂tHze+ +∞ Z −∞ λez(ξ)dν(ξ), ∂tλµ`+ ωµ`2 pµ`= Hz(1), ∂tpµ` = λµ`, ` = 0, . . . , nµ. (57)

The equivalence result of this system to the system (51) mimics the result of Proposition 2.15.

Proposition 4.2. Let (Ey(1), Hze) solve the system (56) with the initial conditions chosen as Hze|t=0= Bz|t=0=

Hz0 and as zero for the rest of unknowns. Let Ey, Hz be the solution to the system (51) with the initial

conditions chosen as Hz|t=0= Bz|t=0= Hz0 and as zero for the rest of unknowns. Then E (1)

y (t) = Ey(t) and

Be

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Similarly, the energy conservation is a corollary of Proposition 3.3, since (56) is a particular case of (48). Proposition 4.3. An energy E associated with (56), defined below, satisfies dtdE = 0. Here

E = 1 2  kE(1) y k 2+ kHe zk 2+ Ee y+ E e z, E e α= 1 2 ∞ Z −∞ kλe αk 2+ ξ2kpe αk 2 dν(ξ), α ∈ {x, y}.

4.2.2. The PML System and Its Stability

Let us now apply the PMLs to the time-domain system (56), or, in the Laplace domain (55). Performing the change of variables (9) with ψx = ε−1y results in the following system (recall that Hz0 is supported outside of

the perfectly matched layer) inside the perfectly matched layer:

q (sµ)(sεy) ˆEy(1)  1 +σx(x) sεy  = −∂xHˆze, q (sµ)(sεy) ˆHze  1 +σx(x) sεy  = −∂xEˆy(1),

or, alternatively, in the whole domain: q (sµ)(sεy) ˆEy(1)+ σx(x) q (sµ)(sεy)−1Eˆy(1)= −∂xHˆze, q (sµ)(sεy) ˆHze+ σx(x) q (sµ)(sεy)−1Hˆze− Hz0= −∂xEˆ(1)y . (58)

To write the above in the time domain, we again apply Lemma 3.2 to the passive function s−1p(sεy)−1(sµ),

with the use of lim

sr→+∞

s−1p(sεy)−1(sµ) = 0. This results in the following representation:

s−1 q (sεy)−1(sµ) = +∞ Z −∞ dν∗(ξ) s2+ ξ2. (59)

With the help of the above we can write the system (58) in the time domain. We make use of s(s2+ ξ2)−1Eˆ(1) y =

ˆ

λey and a similar identity for ˆλez, ˆHze:

∂tDye+ σx(x) +∞ Z −∞ λey(ξ)dν∗(ξ) = −∂xHze, ∂tBze+ σx(x) +∞ Z −∞ λez(ξ)dν∗(ξ) = −∂xEy(1), ∂tDye= ∂tEy(1)+ +∞ Z −∞ λey(ξ)dν(ξ), ∂tλey+ ξ 2pe y= E (1) y , ∂tpey= λ e y, ∂tBze= ∂tHze+ +∞ Z −∞ λez(ξ)dν(ξ), ∂tλez+ ξ2pze= Hze, ∂tpez= λez. (60)

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Theorem 4.4. Let σx(x) ∈ L∞(R). An energy of the PML system (60) defined by E = kE(1) y k 2+ kHe zk 2+ E y+ Ez+ Ey∗+ Ez∗, Eαe = +∞ Z −∞ kλeαk + ξ 2 kpeαk 2 dν(ξ), Eα∗= +∞ Z −∞ Z R |λeα| 2 + ξ2|peα| 2 σ x(x)dxdν∗(ξ), α ∈ {x, z}, remains constant: d dtE = 0.

Proof. The result is obtained by testing the first equation of (60) with Ey(1), the second equation with Hze and

summing up the two obtained identities. The details are left to the reader.  First of all, in the above we can clearly see the role of the condition σx(x) ≥ 0: it is required for the stability of

the PML system. Unlike the results of previous sections for σx≡ const, the above result shows the conservation

of a certain energy, rather than its decay. In practice this may appear non-optimal.

The non-optimality becomes clear when one considers the case µ(s) = εy(s), for which one can obtain a finer

energy decay result in Section 4.1, but for which the results of Theorem 4.4 still remain valid. For instance, in this case, (59) degenerates to a very special representation of the symbol s−1, namely s−1= π−1

+∞

R

−∞ dξ

s22, s ∈ C+.

In this case, on one handp(sεy)−1(sµ) ˆEy = ˆEyin the time domain corresponds to Ey, and on the other hand

it can be represented as π−1

+∞

R

−∞

λeydξ, see (60).

Nonetheless, the energy conservation result obtained in this section is not surprising, and is a consequence of the fact that the system (60) can be viewed as a conservative extension of an equivalent PML system, see [14].

As for the original unknown Hz(1), the corresponding stability result mimics Proposition 2.17.

Proposition 4.5. For the solution of the system (60) coupled with (57), the following holds true:

˜ Ez(t) := 1 2 kH (1) z (t) − H e z(t)k 2+ nµ X `=0 µ`  kλ(1)µ`(t)k2+ ω2µ`kp(1)µ`(t)k2 ! ≤ 2 ˜Ez(0) + CE (0)t2,

where E (t) is defined in Theorem 4.4.

Proof. See the proof of Proposition 2.17 in Appendix A.  The above result indicates a possible linear growth of the field Hz(1). At the moment it is not clear whether

this estimate on the behaviour of the field Hz(1) is optimal, cf. Remark 2.18.

5. Conclusions and Open Questions

In this work we have examined the stability of the PMLs of [5] for dispersive models with the help of the energy techniques. We exploit the coercivity of the corresponding sesquilinear form in the Laplace domain, and based on its analysis, we show how a set of auxiliary PML unknowns can be introduced to simplify the derivation of the energy estimates. This technique allows to obtain the stability result for a constant absorption parameter in 2D, and can be extended to the analysis of the PML with a non-constant absorption parameter in 1D.

There are many open questions remaining, among them the derivation of the energy estimates for a non-constant absorption parameter in two dimensions, even for classical PMLs and non-dispersive case. Another interesting question is whether the technique of this work can be extended to analyze the stability of classical PMLs in 3D. To our knowledge, no energy estimates are known in this case even for a constant absorption parameter. This constitutes a subject of a future research.

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Acknowledgements

I am grateful to Patrick Joly for many useful suggestions and remarks on this work.

Appendix A. Proof of Proposition 2.17

Proof. Testing (42) with Hz(1)− Hze, we obtain

 ∂t(Hz(1)− H e z), H (1) z − H e z  + nµ X `=0 µ`  λ(1)µ`, Hz(1)= nµ X `=0 µ`  λ(1)µ`, Hze+ ne µ X `=0 µeλeµ`, Hz(1)− He z  . (61)

The left-hand side of the above is dtdE˜z, see also (7). As for the right-hand side, let us consider each term

separately. The first term, with the use of the Cauchy-Schwartz formula, can be bounded nµ X `=0 µ`  λ(1)µ`, Hze ≤ kHe zk nµ X `=0 µ` λ (1) µ` ≤ CkH e zk nµ X `=0 µ`kλ (1) µ`k 2 !12 ≤ C0 q E(0) ˜Ez, C0> 0,

where we used the energy identity of Theorem 2.19 and the definition of ˜Ez. Similarly, for some C0 > 0, the

second term of the right-hand side of (61) satisfies ne µ X `=0 µeλeµ`, Hz(1)− He z  ≤ C0kH(1) z − H e zk ne µ X `=0 µekλe µ`k.

To bound the above, we notice that Theorem 2.19 implies that kλe

µ`(t)k ≤ C

E(0), with t ≥ 0 and C

indepen-dent of t, see also Remark 2.14. Thus, for some constant C1> 0,

ne µ X `=0 µeλeµ`, Hz(1)− He z  ≤ C1 q E(0) ˜Ez.

Thus we get from (61) the following differential inequality: d dt ˜ Ez≤ C2 q E(0) ˜Ez, C2> 0, or 1 2 d dt q ˜ Ez≤ C2pE(0).

From the above we obtain

˜ Ez(t) ≤  2C2pE(0)t + q ˜ Ez(0) 2 ≤ 24C22E(0)t2+ ˜Ez(0)  .  The above result shows that kHz(1)(t)−Hze(t)k grows at most linearly, and since kHze(t)k is uniformly bounded

(28)

References

[1] J.-P. B´erenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114 (2) (1994) 185–200. [2] J.-P. B´erenger, Three-dimensional perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys.

127 (2) (1996) 363–379.

[3] E. B´ecache, S. Fauqueux, P. Joly, Stability of perfectly matched layers, group velocities and anisotropic waves, J. Comput. Phys. 188 (2) (2003) 399–433.

[4] E. B´ecache, P. Joly, V. Vinoles, On the analysis of perfectly matched layers for a class of dispersive media. application to negative index metamaterials. Submitted .

URL https://hal.inria.fr/hal-01327315

[5] E. B´ecache, M. Kachanovska, Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: Necessary and sufficient conditions of stability. Submitted.

URL https://hal.archives-ouvertes.fr/hal-01356811

[6] A. Welters, Y. Avniel, S. G. Johnson, Speed-of-light limitations in passive linear media, Phys. Rev. A 90 (2014) 023847. [7] S. A. Cummer, Perfectly matched layer behavior in negative refractive index materials, IEEE Antennas and Wireless

Propa-gation Letters 3 (1) (2004) 172–175.

[8] E. B´ecache, P. Joly, On the analysis of B´erenger’s perfectly matched layers for Maxwell’s equations, M2AN Math. Model. Numer. Anal. 36 (1) (2002) 87–119.

[9] L. Zhao, A. C. Cangellaris, Gt-pml: generalized theory of perfectly matched layers and its application to the reflectionless truncation of finite-difference time-domain grids, IEEE Transactions on Microwave Theory and Techniques 44 (12) (1996) 2555–2563.

[10] T. Hagstrom, D. Appel¨o, Automatic symmetrization and energy estimates using local operators for partial differential equa-tions, Comm. Partial Differential Equations 32 (7-9) (2007) 1129–1145.

[11] D. Appel¨o, T. Hagstrom, G. Kreiss, Perfectly matched layers for hyperbolic systems: general formulation, well-posedness, and stability, SIAM J. Appl. Math. 67 (1) (2006) 1–23.

[12] E. B´ecache, P. Joly, M. Kachanovska, Stable perfectly matched layers for a cold plasma in a strong background magnetic field. Submitted.

URL https://hal.inria.fr/hal-01397581v1

[13] ´E. B´ecache, P. Joly, M. Kachanovska, V. Vinoles, Perfectly matched layers in negative index metamaterials and plasmas, in: CANUM 2014—42e Congr`es National d’Analyse Num´erique, Vol. 50 of ESAIM Proc. Surveys, EDP Sci., Les Ulis, 2015, pp. 113–132.

[14] M. Cassier, P. Joly, M. Kachanovska, On properties of passive systems In preparation.

[15] E. B´ecache, M. Kachanovska, Stable perfectly matched layers for a class of anisotropic dispersive models. Part I. Necessary and Sufficient Conditions of Stability. Extended Version.

URL https://hal.archives-ouvertes.fr/hal-01356811

[16] J. Diaz, P. Joly, A time domain analysis of PML models in acoustics, Comput. Methods Appl. Mech. Engrg. 195 (29-32) (2006) 3820–3853.

[17] E. B´ecache, P. G. Petropoulos, S. D. Gedney, On the long-time behavior of unsplit perfectly matched layers, IEEE Trans. Antennas and Propagation 52 (5) (2004) 1335–1342.

[18] S. Ervedoza, E. Zuazua, Perfectly matched layers in 1-d: energy decay for continuous and semi-discrete waves, Numer. Math. 109 (4) (2008) 597–634.

[19] Y. Huang, J. Li, W. Yang, Mathematical analysis of a PML model obtained with stretched coordinates and its application to backward wave propagation in metamaterials, Numer. Methods Partial Differential Equations 30 (5) (2014) 1558–1574. doi:10.1002/num.21824.

URL http://dx.doi.org/10.1002/num.21824

[20] M. Kachanovska, A note on the stability of PMLs for a class of anisotropic dispersive models in corners. In preparation . [21] A. Bamberger, B. Cockburn, Y. Goldman, M. Kern, P. Joly, Numerical solution of Maxwell’s equations in a conductive and

polarizable medium, in: Proceedings of the Eighth International Conference on Computing Methods in Applied Sciences and Engineering (Versailles, 1987), Vol. 75, 1989, pp. 11–25.

[22] B. Cockburn, P. Joly, Maxwell equations in polarizable media, SIAM Journal on Mathematical Analysis 19 (6) (1988) 1372– 1390.

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